Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-y^3 + y^2 + 72y}{7y^2 - 98y + 315}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-y(y^2 - y - 72)} {7(y^2 - 14y + 45)} $ $ p = -\dfrac{y}{7} \cdot \dfrac{y^2 - y - 72}{y^2 - 14y + 45} $ Next factor the numerator and denominator. $ p = - \dfrac{y}{7} \cdot \dfrac{(y - 9)(y + 8)}{(y - 9)(y - 5)}$ Assuming $y \neq 9$ , we can cancel the $y - 9$ $ p = - \dfrac{y}{7} \cdot \dfrac{y + 8}{y - 5}$ Therefore: $ p = \dfrac{ -y(y + 8)}{ 7(y - 5)}$, $y \neq 9$